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Obtaining the Spatial Distribution of Water Content along a TDR Probe Using the SCEM-UA Bayesian Inverse Modeling Scheme

^a Royal Haskoning, P.O. Box 8520, 3009 AM Rotterdam, The Netherlands
^b Institute for Biodiversity and Ecosystem Dynamics (IBED), University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands
^c Institute for Landscape Ecology and Resource Management, Justus Liebig University, Heinrich-Buff-Ring 26-32, D-35392 Giessen, Germany

View larger version (15K): [in a new window]   Fig. 1. High-resolution 251-point waveform (circles) and low-resolution 1024-point waveform (crosses). The arrows in the inset mark waveform features missed by the 1024-point waveform.

View larger version (11K): [in a new window]   Fig. 2. Overview of the multisection transmission line connected to the cable tester. The numbers indicate the transmission-line section numbering convention adopted for this paper following the approach of Feng et al. (1999). Section 4 is the transmission-line section inside the cable tester, Section 3 is the coaxial cable connecting the probe to the CT, Section 2 is the probe to cable interface, and finally Section 1 is the probe.

View larger version (20K): [in a new window]   Fig. 3. Measured and simulated input signals based on optimized parameters. The top chart shows a small section of the input signal in the time domain. The middle and bottom charts show the magnitude and phase of the complex input signal as a function of the complete frequency range. Solid lines show the results for the model; the open circles are points in the measured waveform.

View larger version (17K): [in a new window]   Fig. 4. Waveforms from open-end cables used to optimize the parameters for the internal transmission-line section of the cable tester and the RG58-C/U coaxial cable. Open circles indicate measurements; solid lines are model fits.

View larger version (14K): [in a new window]   Fig. 5. Measured and modeled waveforms in air and water after calibration. The solid lines are the results from the four-section model; open circles are the sampled points of the waveform sections. The top chart is of the high-resolution 251-point measurements with optimized cable-tester resolution. The bottom chart shows the 1024-point measurements with fixed cable-tester resolution.

View larger version (34K): [in a new window]   Fig. 6. Plots of measured (red circles) and simulated (lines) waveforms of a probe submerged in water to different depths. The top chart is of the 251-point waveform with optimized cable-tester resolution; the bottom chart shows the 1024-point measurements with fixed cable-tester resolution. The waveforms marked with the letter a are difficult to analyze with the algorithm by Heimovaara and Bouten (1990). The first reflection of waveform marked with the letter t is not identified correctly by the tangent analysis method.

View larger version (9K): [in a new window]   Fig. 7. Plot of the optimized insertion depth in water against the measured mean insertion depth of the mean length of the three wires for Exp. 1.

View larger version (11K): [in a new window]   Fig. 8. Linear regression of estimated travel-time from the parameters obtained with SCEM-UA against the travel time obtained from the two-point tangent algorithm of Heimovaara and Bouten (1990).

View larger version (9K): [in a new window]   Fig. 9. Plot of the standard deviations calculated from the distributions of the optimized insertion depth in water obtained with the SCEM-UA algorithm for Exp. 1.

View larger version (13K): [in a new window]   Fig. 10. Plots of the measured waveforms in the two packed samples. The order of packing is marked in the plot and is from probe head to probe end.

View larger version (18K): [in a new window]   Fig. 11. Comparison between measured and optimized waveforms for the two packed samples.

View larger version (12K): [in a new window]   Fig. 12. Range of the real part of the dielectric permittivity between 100 MHz and 2 GHz along the probe, calculated from the optimized parameters for the two packed samples. The probe is divided into two sections in the two-layer model. For the four- and eight-layer models the number of sections was doubled by halving the lengths of the two- and four-layer models.

View larger version (11K): [in a new window]   Fig. 13. Ninety-percent confidence interval for the water content distribution in the packed samples for the two-, four-, and eight-layer models. Probability was calculated by sampling the optimized parameter distributions.